Sen. Ted Cruz of Texas on Friday hit back at Senate Minority Leader Charles E. Schumer for labeling his latest health proposal a “hoax,” saying Democrats made a series of promises about their signature health program that have fallen flat.

“Chuck Schumer knows a lot about health care hoaxes — because the real hoax is the one he and Barack Obama sold to the American people when they first rammed through Obamacare,” Mr. Cruz, Texas Republican, said in a lengthy retort, accusing Democrats of issuing a “litany of lies” to push their law through in 2010.

He pointed to Americans who lost plans that didn’t meet the Affordable Care Act’s coverage requirements, the fact that premiums are rising instead of falling on Obamacare’s exchanges and, in “the ultimate insult,” a policy architect’s claim that Democrats relied on the “stupidity of the American voter” to get their law through Congress.

Republicans who grabbed every lever of elected political power in November are struggling to replace Obamacare with something better, however.

The Congressional Budget Office is expected to score an amendment by Mr. Cruz that would let insurers offer plans that do not comply with Obamacare’s coverage regulations so long as they offer at least one plan that does.

Conservatives say allowing consumers to buy the type of coverage they want is the best way to drive down premiums, though opponents say consumers will pay less for skimpier benefits, while those who still want or need the type of robust coverage mandated by Obamacare will have to pay more.

Mr. Schumer, New York Democrat, issued a statement Thursday designed to blunt any momentum around the idea. He said sicker Americans would be segmented to a pool with ever-rising costs since healthier people will gravitate toward the cheaper, bare-bones plans.

“Americans with pre-existing conditions will almost certainly be left without access to affordable and quality health care, making this even worse than the House bill on this issue,” Mr. Schumer said. “The only way to truly improve our nation’s health care system is for the Trump administration to stop sabotaging the marketplaces and for Republicans to finally heed Democrats’ requests to come together and work in a bipartisan way.”

Conservatives are warming to Mr. Cruz’s idea, saying if Republicans refuse to deliver an outright repeal bill, it would be the next best thing.

Moderate Republicans might balk if the plan is seen as raising costs for people with pre-existing conditions, especially if the CBO comes back with an ugly score.

Mr. Cruz said the exchanges have “very significant subsidies,” and Republicans are offer stabilization funds to offset costs that might rise from segmenting the market.

“Far better to have it through direct tax revenue and let the people who are struggling pay a much more affordable premium,” Mr. Cruz told reporters.

Senate Majority Leader Mitch McConnell signaled Thursday that Congress will have to patch up Obamacare’s ailing markets if Republicans cannot smooth over their differences and pass a replacement plan.